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Research ArticleVibration Analysis of Steel-Concrete Composite BoxBeams considering Shear Lag and Slip
Zhou Wangbao1 Li Shu-jin1 Jiang Lizhong2 and Qin Shiqiang1
1School of Civil Engineering and Architecture Wuhan University of Technology Wuhan 430070 China2School of Civil Engineering Central South University Changsha 410075 China
Correspondence should be addressed to Zhou Wangbao zhwbwhuteducn
Received 21 January 2015 Revised 15 March 2015 Accepted 15 March 2015
Academic Editor Giovanni Garcea
Copyright copy 2015 Zhou Wangbao et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In order to investigate dynamic characteristics of steel-concrete composite box beams a longitudinal warping function of beamsection considering self-balancing of axial forces is established On the basis of Hamilton principle governing differential equationsof vibration and displacement boundary conditions are deduced by taking into account coupled influencing of shear lag interfaceslip and shear deformationTheproposedmethod shows an improvement over previous calculationsThe central differencemethodis applied to solve the differential equations to obtain dynamic responses of composite beams subjected to arbitrarily distributedloads The results from the proposed method are found to be in good agreement with those from ANSYS through numericalstudies Its validity is thus verified and meaningful conclusions for engineering design can be drawn as follows There are obviousshear lag effects in the top concrete slab and bottom plate of steel beams under dynamic excitation This shear lag increases withthe increasing degree of shear connections However it has little impact on the period and deflection amplitude of vibration ofcomposite box beamsThe amplitude of deflection and strains in concrete slab reduce as the degree of shear connections increasesNevertheless the influence of shear connections on the period of vibration is not distinct
1 Introduction
The advantages of fully making use of compressive strengthof concrete and tensile strength of steel make composite boxbeams popular in the bridge engineering For composite boxbeams with large flange width however there are shear lageffects in the top concrete slab and bottom plates of steel boxbeams due to the nonuniform transverse distribution of shearstress across section In addition due to the fact that the shearconnectors between the steel beam and concrete slab cannotbe absolutely rigid there exists relative slip between themeven for fully shear connected beamsTherefore the behaviorof steel-concrete composite box beams suffers from coupledeffects of both the shear lag and shear slip [1ndash5] Gara [6 7]presented a beam finite element in which the warping of theslab cross section was considered for the long-term analysisof steel-concrete composite decks taking into account theshear lag in the slab and the partial shear interaction atthe slab-beam interface By using energy variation method
Zhou et al [5] derived the governing differential equationsand boundary conditions of the steel-concrete composite boxbeams by considering the longitudinal warp caused by shearlag effects and slip between steel beams and concrete slabs
Morassi and Dilena [8ndash10] investigated an experimenton damage-induced changes in modal parameters of steel-concrete composite beams subject to small vibrations andthe experiments revealed that flexural frequencies showeda rather high sensitivity to damage and therefore can beconsidered as a valid indicator upon a diagnostic analysisTherefore it makes sense to study the vibration characteristicof composite beam Adam et al [11] analyzed the dynamicflexural behavior of elastic two-layer beams with interlayerslip by assuming the Bernoulli-Euler hypothesis to hold foreach layer separately and considering a linear constitutiveequation between the horizontal slip and the interlayershear force Biscontin et al [12] performed an experimentalanalytical investigation on the dynamic behavior of I-steel-concrete composite beams subject to small vibrations and a
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 601757 8 pageshttpdxdoiorg1011552015601757
2 Mathematical Problems in Engineering
one-dimensional model of a composite beam was presentedwhere the elements connecting the steel and reinforced con-crete slab were described by means of a strain energy densityfunction defined along the beam axis Berczynski [13 14] pre-sented a solution of the problem of free vibrations of I-steel-concrete composite beams and found that the results obtainedon the basis of the Timoshenko beam theory model achievedthe highest conformitywith the experimental results both forhigher and for lowermodes of flexural vibrations of the beamXu andWu [15] investigated the static dynamic and bucklingbehavior of partial-interaction T-composite members bytaking into account the influences of rotary inertia andshear deformations and obtained the analytical expressionsof the frequencies of the simply supported composite beamShen et al [16] studied the dynamic behavior of partial-interaction T-composite beams by using state-space methodwhich was properly established via selecting the appropriatestate variables and the characteristic equations of frequencyand the corresponding modal shapes of free vibration undergeneralized boundary conditions were then obtained Shenand Zhong [17] examined the deformation of partially I-steel-concrete composite beams under distributed loading andfree vibrations of partially I-steel-concrete composite beamsunder various boundary conditions where the weak-formquadrature element method was used
Considerable efforts have been put into the investigationof dynamic characteristics of composite beams includinginterface slip effects [18ndash22] Most of studies focus on the I-steel beam-concrete composite beams or T-type compositebeams The studies about the dynamic characteristics ofcomposite box beams are rare especially the studies involvingboth the interface slip and shear lag in the dynamic character-istics Based on a longitudinal warping function consideringself-balancing of axial forces and the Hamilton principlethis paper deduces the governing differential equations ofdynamic responses of composite box beams under arbitrarilydistributed loads It takes into account the influence of shearlag interface slip and shear deformation The equations aresolved by central difference methods Numerical studies arecarried out and a good agreement is achieved between resultsfrom the proposed method and finite element method usingANSYS The influences of the shear lag effect and the degreeof shear connections on the dynamic responses of compositebox beams are examined and meaningful conclusions forengineering design are drawn
2 Basic Assumptions
Figure 1 shows the section size and coordinate system ofcomposite box beams The parameters 119905
1 1199052 1199053 1199054 and
119905lowast
4are the thicknesses of the top concrete slab cantilever
plate bottom plate web and top flange of steel box beamrespectively the parameters 119887
1 1198872 1198873 1198874 and 119887
lowast
4are the
widths of half-top concrete slab cantilever plate half-bottomplate of steel beams web of steel beams and top flange ofsteel beams respectively the parameters ℎ
119888and ℎ
119904are the
distances between the centroids of the concrete slab andsteel beam to the slab-beam interface respectively and here
hc
hsy 0
z w
b1b2 b1 b2
b4
blowast
4
t4
t1
b3 b3
t3
tlowast
4h
t2
Figure 1 Schematic of section of steel-concrete composite boxbeams
ℎ = ℎ119888+ ℎ119904 Reasonable assumptions to simplify the analysis
model are made as follows(1) According to the displacement compatibility the
longitudinal warping function of concrete top slab cantileverplate bottom flange and web of steel beams is assumed as[5 23 24]
119892119894= 120595119894(119910)119880 (119909 119905) 119894 = 1 2 3 4 (1)
120595119894= 120572119894(1199102
1198872
119894
minus 1) + 119889 119894 = 1 2 3 4 (2)
Considering self-balancing of axial forces produced bylongitudinal displacement yields [5 23 24]
1205721= 1 120572
2=
1198872
2
1198872
1
1205723= 119911119887
1198872
3
(1198872
1119911119905) 120572
4= 0
(3)
Considering self-balancing of axial forces produced bylongitudinal warping function yields [5 23 24]
int
119860
120595119889119860 = 0 (4)
Substituting (2) into (4) obtains the constant term oflongitudinal warping displacements as follows
119889 =2 (12057211198601119899 + 120572
21198602119899 + 120572
31198603)
(31198600)
(5)
For 119894 = 2 replace 119910 with 119910 = 1198871+ 1198872minus 119910 119911
119905and
119911119887are the 119911-coordinates of centroids of the concrete slab
and bottom flange 1199110is the 119911-coordinate of the slab-beam
interface 119889 is the constant term of longitudinal warpingdisplacements 119880(119909 119905) is the function of the amplitude ofwarping displacements 120595
119894is the warping shape function of
the beam section as shown in Figure 2120572119894is the self-balancing
coefficient of section warping 119899 = 119864119904119864119888 and 119864
119904and 119864
119888are
the modulus of elasticity of steel and concrete respectively
1198601= 211988711199051 119860
2= 211988721199052
1198603= 211988731199053 119860
4= 2 (119887
41199054+ 119887lowast
4119905lowast
4)
1198600=
(1198601+ 1198602)
119899+ 1198603+ 1198604
(6)
Mathematical Problems in Engineering 3
A998400
1A998400
2
A998400
3
d
Figure 2 Schematic of warping shape function
(2) The longitudinal displacement of any point in thetransverse section of composite box beams is assumed as thesuperposition of the longitudinal displacement based on theplain section assumption the longitudinal warping displace-ment due to the shear lag and longitudinal displacement dueto the relative interface slip It can be expressed as [5 23 24]
119906119894= 119896119888120585 minus (119911 minus 119911
119905) 120579 + 119892
119894119894 = 1 2 (7)
119906119894= 119896119904120585 minus (119911 minus 119911
119904) 120579 + 119892
119894119894 = 3 4 (8)
119896119888= minus
119860119904
1198600
119896119904=
119860119888
(1198991198600) (9)
120577 (119909 119905) = 120585 + ℎ120579 (10)
where 120579(119909 119905) is the rotation of the beam section 119860119908= 211988741199054
119860119904= 1198603+ 1198604is the cross section area of steel beams 119860
119888=
1198601+ 1198602is the cross section area of concrete slabs 120585(119909 119905) is
the longitudinal displacement difference between centroidsof the concrete slab and steel beam 120577(119909 119905) is slab-beaminterface slip 119911
119904is the 119911-coordinate of the centroid of the
steel beam 119896119904is the ratio between steel beamrsquos longitudinal
displacement due to the relative interface slip and relativeinterface slip and 119896
119888is the ratio between concrete slabrsquos
longitudinal displacement due to the relative interface slipand relative interface slip
(3) The vertical compression and transverse strain ofconcrete slabs and steel beams are ignored [5 23]
3 Vibration Differential Equation andBoundary Conditions
31 The Strain of the Cross Section The sectional strain canbe obtained from the above longitudinal displacement ofcomposite beam sections as
120576119909119894
= 119896119888
120597120585
120597119909minus (119911 minus 119911
119905)120597120579
120597119909+ 120595119894
120597119880
120597119909119894 = 1 2
120576119909119894
= 119896119904
120597120585
120597119909minus (119911 minus 119911
119904)120597120579
120597119909+ 120595119894
120597119880
120597119909119894 = 3 4
120574119909119910119894
=120597119906119894
120597119910=
120597120595119894
120597119910119880 119894 = 1 2 3
120574119909119911
=120597119908
120597119909minus 120579
(11)
where 120576119894(119894 = 1 2 3 4) are the longitudinal strain of the top
concrete slab cantilever plate bottom flange and web of steelbeams respectively 120574
119894(119894 = 1 2 3) are the shear strain of
top concrete slab cantilever plate and bottom plate of steelbeams respectively 120574
119909119911is the shear strain of the web of steel
beams 119908(119909 119905) is the vertical deflection of composite boxbeams
32 Total Potential Energy of the Composite Box Beam Thestrain energy of composite box beams is defined as [5]
119881 = 05 int
119871
0
int
1198600
(1198641199041205762+ 1198661199041205742) 119889119860119889119909
+ 05 int
119871
0
119896sl1205772119889119909 +
051198661199041198600(1199081015840minus 120579)2
120572119904
(12)
Substituting (10)-(11) into (12) gives the strain energy ofcomposite box beams as
119881 = 05 int
119871
0
[
[
11986312058510158402+ 11986511988010158402+ 2119867119880
10158401205851015840+ 11986812057910158402+ 1198691198802
minus 211987811988010158401205791015840+ 119896sl120577
2+
1198661199041198600(1199081015840minus 120579)2
120572119904
]
]
119889119909
(13)
where119863 = 1198641198881198962
119888119860119888+1198641199041198962
119904119860119904 119865 = 119864
119888119861119888119891+119864119904119861119904119891119867 = 119864
119888119861119888ℎ+
119864119904119861119904ℎ 119868 = 119864
119888119861119888119894+119864119904119861119904119894 119869 = 119866
119888119861119888119895+119866119904119861119904119895 119878 = 119864
119888119861119888119904+119864119904119861119904119904
119861119888119891
= int119860119888
1205952119889119860 119861
119904119891= int119860119904
1205952119889119860 119861
119888ℎ= int119860119888
119896119888120595119889119860 119861
119904ℎ=
int119860119904
119896119904120595119889119860 119861
119888119894= int119860119888
(119911 minus 119911119905)2119889119860 119861
119904119894= int119860119904
(119911 minus 119911119904)2119889119860 119861
119888119895=
int119860119888
(120597120595120597119910)2119889119860 119861
119904119895= int1198603
(120597120595120597119910)2119889119860 119861
119888119904= int119860119888
(119911minus119911119905)120595 119889119860
119861119904119904
= int119860119904
(119911 minus 119911119904)120595 119889119860 and 120572
119904is the correction coefficient of
shear deformation Considering that the webs bear most ofthe vertical shear force in section here the value of 120572
119904is taken
as 1198600(211988741199054) and 2119887
41199054is the section area of webs 119896sl is the
slip stiffness between the concrete slab and the steel beam119871 isthe span of the composite box beam 119866
119904is the shear modulus
of the steel beamThe kinetic energy of the composite box beam is [5]
119879 =1
2int
119871
0
int
119860
(1205882+ 119898
2) 119889119860119889119909 (14)
Substituting of (7) and (8) yields
119879
=1
2int
119871
(1198982+ 1198631
1205852+ 11986512+ 21198671
120585 + 1198681
1205792minus 21198781 120579) 119889119909
(15)
where1198631= 1205881198881198962
119888119860119888+1205881199041198962
1199041198601199041198651= 120588119888119861119888119891+1205881199041198611199041198911198671= 120588119888119861119888ℎ+
120588119904119861119904ℎ 1198681= 120588119888119861119888119894+ 120588119904119861119904119894 1198781= 120588119888119861119888119904+ 120588119904119861119904119904119898 = 120588
119888119860119888+ 120588119904119860119904
119860 = 119860119904+ 119860119888 and 120588
119888and 120588119904are the density of concrete slabs
and steel beams respectively
4 Mathematical Problems in Engineering
The work done by the external loads can be expressed as
119882 = int
119871
119902 (119909 119905) 119908 119889119909 (16)
where 119902(119909 119905) is the distribution function of arbitrary load
33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as
11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579
10158401015840minus 1198651 minus 119867
1
120585 + 1198781
120579 = 0 (17)
11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867
1 minus 119863
1
120585 = 0 (18)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (19)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878
1 minus 1198681
120579 = 0 (20)
(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880
10038161003816100381610038161003816
119871
0= 0 (21)
(1198631205851015840+ 119867119880
1015840) 120575120585
10038161003816100381610038161003816
119871
0= 0 (22)
1198661199041198600(1199081015840minus 120579)
120572119904
120575119908
10038161003816100381610038161003816100381610038161003816100381610038161003816
119871
0
= 0 (23)
(1198681205791015840minus 1198781198801015840) 120575120579
10038161003816100381610038161003816
119871
0= 0 (24)
Taking a simply supported beam as example (21)ndash(24)give boundary conditions as
1198801015840(119871 119905) = 120585
1015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1198801015840(0 119905) = 120585
1015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(25)
given the initial conditions as
119880 (119909 0) = 1198800(119909) (119909 0) = 119880
0(119909)
120585 (119909 0) = 1205850(119909)
120585 (119909 0) = 120585
0(119909)
119908 (119909 0) = 1199080(119909) (119909 0) = 119908
0(119909)
120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579
0(119909)
(26)
4 Finite Difference Method ofVibration Differential Equation
41 Difference Scheme Let solution domain be 120590 = (119909 119905) |
0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that
119909119894= 119894120592 (119894 = 0 1 2 119868)
119905119895= 119895120591 (119895 = 0 1 2 119869)
(27)
where 119868 = 119871120592 119869 = 119879120591
Let
119880119895
119894= 119880 (119909
119894 119905119895) 120585
119895
119894= 120585 (119909
119894 119905119895)
119908119895
119894= 119908 (119909
119894 119905119895) 120579
119895
119894= 120579 (119909
119894 119905119895)
(28)
The central difference calculation of governing differen-tial equation yields
119865
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119867
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119878
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 1198651
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 119869119880119895
119894minus 1198671
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
+ 1198781
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(29)
119867
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119863
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119896sl (120585119895
119894+ ℎ120579119895
119894) minus 119867
1
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198631
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(30)
1198661199041198600
120572119904
(
119908119895
119894+1minus 119908119895
119894minus1
2120592minus 120579119895
119894) + 119868
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 119878
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 1198781
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198681
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
minus 119896slℎ (120585119895
119894+ ℎ120579119895
119894) = 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(31)
1198661199041198600
120572119904
(
119908119895
119894+1minus 2119908119895
119894+ 119908119895
119894minus1
1205922
minus
120579119895
119894+1minus 120579119895
119894minus1
2120592)
+ 119902119895
119894minus 119898
119908119895+1
119894minus 2119908119895
119894+ 119908119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(32)
Mathematical Problems in Engineering 5
The difference calculation of boundary conditions yields
119880119895
119868minus 119880119895
119868minus1
120592=
119880119895
1minus 119880119895
0
120592= 0 119895 = 0 1 119869
(33)
120585119895
119868minus 120585119895
119868minus1
120592=
120585119895
1minus 120585119895
0
120592= 0 119895 = 0 1 119869
(34)
119908119895
119868= 119908119895
0= 0 119895 = 0 1 119869 (35)
120579119895
119868minus 120579119895
119868minus1
120592=
120579119895
1minus 120579119895
0
120592= 0 119895 = 0 1 119869
(36)
The difference calculation of initial condition yields
1198800
119894= 1198800119894
1198801
119894minus 1198800
119894
120591= 1198800119894
119894 = 0 1 2 119868
1205850
119894= 1205850119894
1205851
119894minus 1205850
119894
120591= 1205850119894
119894 = 0 1 2 119868
1199080
119894= 1199080119894
1199081
119894minus 1199080
119894
120591= 1199080119894
119894 = 0 1 2 119868
1205790
119894= 1205790119894
1205791
119894minus 1205790
119894
120591= 1205790119894
119894 = 0 1 2 119868
(37)
42 Solution Step Let U119895 = 119880119895
0 119880119895
1 119880
119895
119868119879 120585119895 = 120585
119895
0 120585119895
1
120585119895
119868119879 w119895 = 119908
119895
0 119908119895
1 119908
119895
119868119879 and 120579119895 = 120579
119895
0 120579119895
1 120579
119895
119868119879
Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below
(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)
(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880
119895+1
1 119880119895+1
2 119880
119895+1
119868minus1 120585119895+1
1 120585119895+1
2 120585
119895+1
119868minus1 and
120579119895+1
1 120579119895+1
2 120579
119895+1
119868minus1 from (29)ndash(31)
(c) Calculate 119880119895+1
0 119880119895+1
119868 120585119895+1
0 120585119895+1
119868 and 120579
119895+1
0 120579119895+1
119868
from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)
(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1
1 119908119895+1
2
119908119895+1
119868minus1 from (32)
(e) Calculate 119908119895+1
0 119908119895+1
119868 from (35) and calculate w119895+1
combined with step (d)
43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box
beams and boundary conditions without shear lag effects canbe deduced as
11986312058510158401015840minus 119896sl120577 minus 119863
1
120585 = 0 (38)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (39)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868
1120579 = 0 (40)
1205851015840120575120585
10038161003816100381610038161003816
119871
0= 0 120579
1015840120575120579
10038161003816100381610038161003816
119871
0= 0 (119908
1015840minus 120579) 120575119908
10038161003816100381610038161003816
119871
0= 0 (41)
For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as
1205851015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1205851015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(42)
The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42
5 Analysis of Examples
The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864
119904= 20 times 10
5MPa 119864119888= 45 times 10
4MPa119866119904= 771 times 10
4MPa 120583119904= 028 120583
119888= 018 120588
119904= 7900 kgsdotmminus3
120588119888= 2400 kgsdotmminus3 119887
1= 1198873= 25m 119887
2= 20m 119887
4= 30m
1198875= 02m 119905
1= 1199052= 03m 119905
3= 1199055= 006m 119905
4= 009m
119871 = 30m 119897 = 03m 119899119904= 2 and 119891
119904= 300MPa
The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910
degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4
The 119896sl is calculated as [5 23 24]
119896sl =1198701
119897 (43)
6 Mathematical Problems in Engineering
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
(a) 119903 = 025
000 005 010 015 020 0250
4
8
12
16
20
24
Disp
lace
men
t at m
idsp
an (m
m)
Time (s)
(b) 119903 = 05
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(c) 119903 = 10
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(d) 119903 = 15
Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)
The spring parameter of the COMBIN14 elements can becalculated as [25]
1198701= 066119899
119904119881119906 (44)
The degree of shear connections is calculated as [5]
119903 =119899119904119881119906119871
119860119904119891119904119897 (45)
where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891
119904is the tensile strength of steel beams 119899
119904
is the number of connectors across the beam section119881119906is the
ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from
the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows
(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period
(2) The larger the degree of shear connections thesmaller the amplitude of deflection
(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams
(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects
(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear
Mathematical Problems in Engineering 7
0 1 2 3 4 5100
120
140
160
180
200
minus5 minus4 minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
Stra
in d
istrib
utio
n (10minus6)
(a) Concrete plate
0 1 2 3minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
minus220
minus230
minus240
minus250
minus260
minus270
minus280
Stra
in d
istrib
utio
n (10minus6)
(b) Steel beam bottom plate
Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)
connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection
6 Conclusions
(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams
(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams
(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect
(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)
References
[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003
[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005
[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007
[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014
[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012
[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011
[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
one-dimensional model of a composite beam was presentedwhere the elements connecting the steel and reinforced con-crete slab were described by means of a strain energy densityfunction defined along the beam axis Berczynski [13 14] pre-sented a solution of the problem of free vibrations of I-steel-concrete composite beams and found that the results obtainedon the basis of the Timoshenko beam theory model achievedthe highest conformitywith the experimental results both forhigher and for lowermodes of flexural vibrations of the beamXu andWu [15] investigated the static dynamic and bucklingbehavior of partial-interaction T-composite members bytaking into account the influences of rotary inertia andshear deformations and obtained the analytical expressionsof the frequencies of the simply supported composite beamShen et al [16] studied the dynamic behavior of partial-interaction T-composite beams by using state-space methodwhich was properly established via selecting the appropriatestate variables and the characteristic equations of frequencyand the corresponding modal shapes of free vibration undergeneralized boundary conditions were then obtained Shenand Zhong [17] examined the deformation of partially I-steel-concrete composite beams under distributed loading andfree vibrations of partially I-steel-concrete composite beamsunder various boundary conditions where the weak-formquadrature element method was used
Considerable efforts have been put into the investigationof dynamic characteristics of composite beams includinginterface slip effects [18ndash22] Most of studies focus on the I-steel beam-concrete composite beams or T-type compositebeams The studies about the dynamic characteristics ofcomposite box beams are rare especially the studies involvingboth the interface slip and shear lag in the dynamic character-istics Based on a longitudinal warping function consideringself-balancing of axial forces and the Hamilton principlethis paper deduces the governing differential equations ofdynamic responses of composite box beams under arbitrarilydistributed loads It takes into account the influence of shearlag interface slip and shear deformation The equations aresolved by central difference methods Numerical studies arecarried out and a good agreement is achieved between resultsfrom the proposed method and finite element method usingANSYS The influences of the shear lag effect and the degreeof shear connections on the dynamic responses of compositebox beams are examined and meaningful conclusions forengineering design are drawn
2 Basic Assumptions
Figure 1 shows the section size and coordinate system ofcomposite box beams The parameters 119905
1 1199052 1199053 1199054 and
119905lowast
4are the thicknesses of the top concrete slab cantilever
plate bottom plate web and top flange of steel box beamrespectively the parameters 119887
1 1198872 1198873 1198874 and 119887
lowast
4are the
widths of half-top concrete slab cantilever plate half-bottomplate of steel beams web of steel beams and top flange ofsteel beams respectively the parameters ℎ
119888and ℎ
119904are the
distances between the centroids of the concrete slab andsteel beam to the slab-beam interface respectively and here
hc
hsy 0
z w
b1b2 b1 b2
b4
blowast
4
t4
t1
b3 b3
t3
tlowast
4h
t2
Figure 1 Schematic of section of steel-concrete composite boxbeams
ℎ = ℎ119888+ ℎ119904 Reasonable assumptions to simplify the analysis
model are made as follows(1) According to the displacement compatibility the
longitudinal warping function of concrete top slab cantileverplate bottom flange and web of steel beams is assumed as[5 23 24]
119892119894= 120595119894(119910)119880 (119909 119905) 119894 = 1 2 3 4 (1)
120595119894= 120572119894(1199102
1198872
119894
minus 1) + 119889 119894 = 1 2 3 4 (2)
Considering self-balancing of axial forces produced bylongitudinal displacement yields [5 23 24]
1205721= 1 120572
2=
1198872
2
1198872
1
1205723= 119911119887
1198872
3
(1198872
1119911119905) 120572
4= 0
(3)
Considering self-balancing of axial forces produced bylongitudinal warping function yields [5 23 24]
int
119860
120595119889119860 = 0 (4)
Substituting (2) into (4) obtains the constant term oflongitudinal warping displacements as follows
119889 =2 (12057211198601119899 + 120572
21198602119899 + 120572
31198603)
(31198600)
(5)
For 119894 = 2 replace 119910 with 119910 = 1198871+ 1198872minus 119910 119911
119905and
119911119887are the 119911-coordinates of centroids of the concrete slab
and bottom flange 1199110is the 119911-coordinate of the slab-beam
interface 119889 is the constant term of longitudinal warpingdisplacements 119880(119909 119905) is the function of the amplitude ofwarping displacements 120595
119894is the warping shape function of
the beam section as shown in Figure 2120572119894is the self-balancing
coefficient of section warping 119899 = 119864119904119864119888 and 119864
119904and 119864
119888are
the modulus of elasticity of steel and concrete respectively
1198601= 211988711199051 119860
2= 211988721199052
1198603= 211988731199053 119860
4= 2 (119887
41199054+ 119887lowast
4119905lowast
4)
1198600=
(1198601+ 1198602)
119899+ 1198603+ 1198604
(6)
Mathematical Problems in Engineering 3
A998400
1A998400
2
A998400
3
d
Figure 2 Schematic of warping shape function
(2) The longitudinal displacement of any point in thetransverse section of composite box beams is assumed as thesuperposition of the longitudinal displacement based on theplain section assumption the longitudinal warping displace-ment due to the shear lag and longitudinal displacement dueto the relative interface slip It can be expressed as [5 23 24]
119906119894= 119896119888120585 minus (119911 minus 119911
119905) 120579 + 119892
119894119894 = 1 2 (7)
119906119894= 119896119904120585 minus (119911 minus 119911
119904) 120579 + 119892
119894119894 = 3 4 (8)
119896119888= minus
119860119904
1198600
119896119904=
119860119888
(1198991198600) (9)
120577 (119909 119905) = 120585 + ℎ120579 (10)
where 120579(119909 119905) is the rotation of the beam section 119860119908= 211988741199054
119860119904= 1198603+ 1198604is the cross section area of steel beams 119860
119888=
1198601+ 1198602is the cross section area of concrete slabs 120585(119909 119905) is
the longitudinal displacement difference between centroidsof the concrete slab and steel beam 120577(119909 119905) is slab-beaminterface slip 119911
119904is the 119911-coordinate of the centroid of the
steel beam 119896119904is the ratio between steel beamrsquos longitudinal
displacement due to the relative interface slip and relativeinterface slip and 119896
119888is the ratio between concrete slabrsquos
longitudinal displacement due to the relative interface slipand relative interface slip
(3) The vertical compression and transverse strain ofconcrete slabs and steel beams are ignored [5 23]
3 Vibration Differential Equation andBoundary Conditions
31 The Strain of the Cross Section The sectional strain canbe obtained from the above longitudinal displacement ofcomposite beam sections as
120576119909119894
= 119896119888
120597120585
120597119909minus (119911 minus 119911
119905)120597120579
120597119909+ 120595119894
120597119880
120597119909119894 = 1 2
120576119909119894
= 119896119904
120597120585
120597119909minus (119911 minus 119911
119904)120597120579
120597119909+ 120595119894
120597119880
120597119909119894 = 3 4
120574119909119910119894
=120597119906119894
120597119910=
120597120595119894
120597119910119880 119894 = 1 2 3
120574119909119911
=120597119908
120597119909minus 120579
(11)
where 120576119894(119894 = 1 2 3 4) are the longitudinal strain of the top
concrete slab cantilever plate bottom flange and web of steelbeams respectively 120574
119894(119894 = 1 2 3) are the shear strain of
top concrete slab cantilever plate and bottom plate of steelbeams respectively 120574
119909119911is the shear strain of the web of steel
beams 119908(119909 119905) is the vertical deflection of composite boxbeams
32 Total Potential Energy of the Composite Box Beam Thestrain energy of composite box beams is defined as [5]
119881 = 05 int
119871
0
int
1198600
(1198641199041205762+ 1198661199041205742) 119889119860119889119909
+ 05 int
119871
0
119896sl1205772119889119909 +
051198661199041198600(1199081015840minus 120579)2
120572119904
(12)
Substituting (10)-(11) into (12) gives the strain energy ofcomposite box beams as
119881 = 05 int
119871
0
[
[
11986312058510158402+ 11986511988010158402+ 2119867119880
10158401205851015840+ 11986812057910158402+ 1198691198802
minus 211987811988010158401205791015840+ 119896sl120577
2+
1198661199041198600(1199081015840minus 120579)2
120572119904
]
]
119889119909
(13)
where119863 = 1198641198881198962
119888119860119888+1198641199041198962
119904119860119904 119865 = 119864
119888119861119888119891+119864119904119861119904119891119867 = 119864
119888119861119888ℎ+
119864119904119861119904ℎ 119868 = 119864
119888119861119888119894+119864119904119861119904119894 119869 = 119866
119888119861119888119895+119866119904119861119904119895 119878 = 119864
119888119861119888119904+119864119904119861119904119904
119861119888119891
= int119860119888
1205952119889119860 119861
119904119891= int119860119904
1205952119889119860 119861
119888ℎ= int119860119888
119896119888120595119889119860 119861
119904ℎ=
int119860119904
119896119904120595119889119860 119861
119888119894= int119860119888
(119911 minus 119911119905)2119889119860 119861
119904119894= int119860119904
(119911 minus 119911119904)2119889119860 119861
119888119895=
int119860119888
(120597120595120597119910)2119889119860 119861
119904119895= int1198603
(120597120595120597119910)2119889119860 119861
119888119904= int119860119888
(119911minus119911119905)120595 119889119860
119861119904119904
= int119860119904
(119911 minus 119911119904)120595 119889119860 and 120572
119904is the correction coefficient of
shear deformation Considering that the webs bear most ofthe vertical shear force in section here the value of 120572
119904is taken
as 1198600(211988741199054) and 2119887
41199054is the section area of webs 119896sl is the
slip stiffness between the concrete slab and the steel beam119871 isthe span of the composite box beam 119866
119904is the shear modulus
of the steel beamThe kinetic energy of the composite box beam is [5]
119879 =1
2int
119871
0
int
119860
(1205882+ 119898
2) 119889119860119889119909 (14)
Substituting of (7) and (8) yields
119879
=1
2int
119871
(1198982+ 1198631
1205852+ 11986512+ 21198671
120585 + 1198681
1205792minus 21198781 120579) 119889119909
(15)
where1198631= 1205881198881198962
119888119860119888+1205881199041198962
1199041198601199041198651= 120588119888119861119888119891+1205881199041198611199041198911198671= 120588119888119861119888ℎ+
120588119904119861119904ℎ 1198681= 120588119888119861119888119894+ 120588119904119861119904119894 1198781= 120588119888119861119888119904+ 120588119904119861119904119904119898 = 120588
119888119860119888+ 120588119904119860119904
119860 = 119860119904+ 119860119888 and 120588
119888and 120588119904are the density of concrete slabs
and steel beams respectively
4 Mathematical Problems in Engineering
The work done by the external loads can be expressed as
119882 = int
119871
119902 (119909 119905) 119908 119889119909 (16)
where 119902(119909 119905) is the distribution function of arbitrary load
33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as
11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579
10158401015840minus 1198651 minus 119867
1
120585 + 1198781
120579 = 0 (17)
11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867
1 minus 119863
1
120585 = 0 (18)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (19)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878
1 minus 1198681
120579 = 0 (20)
(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880
10038161003816100381610038161003816
119871
0= 0 (21)
(1198631205851015840+ 119867119880
1015840) 120575120585
10038161003816100381610038161003816
119871
0= 0 (22)
1198661199041198600(1199081015840minus 120579)
120572119904
120575119908
10038161003816100381610038161003816100381610038161003816100381610038161003816
119871
0
= 0 (23)
(1198681205791015840minus 1198781198801015840) 120575120579
10038161003816100381610038161003816
119871
0= 0 (24)
Taking a simply supported beam as example (21)ndash(24)give boundary conditions as
1198801015840(119871 119905) = 120585
1015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1198801015840(0 119905) = 120585
1015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(25)
given the initial conditions as
119880 (119909 0) = 1198800(119909) (119909 0) = 119880
0(119909)
120585 (119909 0) = 1205850(119909)
120585 (119909 0) = 120585
0(119909)
119908 (119909 0) = 1199080(119909) (119909 0) = 119908
0(119909)
120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579
0(119909)
(26)
4 Finite Difference Method ofVibration Differential Equation
41 Difference Scheme Let solution domain be 120590 = (119909 119905) |
0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that
119909119894= 119894120592 (119894 = 0 1 2 119868)
119905119895= 119895120591 (119895 = 0 1 2 119869)
(27)
where 119868 = 119871120592 119869 = 119879120591
Let
119880119895
119894= 119880 (119909
119894 119905119895) 120585
119895
119894= 120585 (119909
119894 119905119895)
119908119895
119894= 119908 (119909
119894 119905119895) 120579
119895
119894= 120579 (119909
119894 119905119895)
(28)
The central difference calculation of governing differen-tial equation yields
119865
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119867
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119878
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 1198651
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 119869119880119895
119894minus 1198671
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
+ 1198781
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(29)
119867
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119863
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119896sl (120585119895
119894+ ℎ120579119895
119894) minus 119867
1
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198631
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(30)
1198661199041198600
120572119904
(
119908119895
119894+1minus 119908119895
119894minus1
2120592minus 120579119895
119894) + 119868
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 119878
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 1198781
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198681
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
minus 119896slℎ (120585119895
119894+ ℎ120579119895
119894) = 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(31)
1198661199041198600
120572119904
(
119908119895
119894+1minus 2119908119895
119894+ 119908119895
119894minus1
1205922
minus
120579119895
119894+1minus 120579119895
119894minus1
2120592)
+ 119902119895
119894minus 119898
119908119895+1
119894minus 2119908119895
119894+ 119908119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(32)
Mathematical Problems in Engineering 5
The difference calculation of boundary conditions yields
119880119895
119868minus 119880119895
119868minus1
120592=
119880119895
1minus 119880119895
0
120592= 0 119895 = 0 1 119869
(33)
120585119895
119868minus 120585119895
119868minus1
120592=
120585119895
1minus 120585119895
0
120592= 0 119895 = 0 1 119869
(34)
119908119895
119868= 119908119895
0= 0 119895 = 0 1 119869 (35)
120579119895
119868minus 120579119895
119868minus1
120592=
120579119895
1minus 120579119895
0
120592= 0 119895 = 0 1 119869
(36)
The difference calculation of initial condition yields
1198800
119894= 1198800119894
1198801
119894minus 1198800
119894
120591= 1198800119894
119894 = 0 1 2 119868
1205850
119894= 1205850119894
1205851
119894minus 1205850
119894
120591= 1205850119894
119894 = 0 1 2 119868
1199080
119894= 1199080119894
1199081
119894minus 1199080
119894
120591= 1199080119894
119894 = 0 1 2 119868
1205790
119894= 1205790119894
1205791
119894minus 1205790
119894
120591= 1205790119894
119894 = 0 1 2 119868
(37)
42 Solution Step Let U119895 = 119880119895
0 119880119895
1 119880
119895
119868119879 120585119895 = 120585
119895
0 120585119895
1
120585119895
119868119879 w119895 = 119908
119895
0 119908119895
1 119908
119895
119868119879 and 120579119895 = 120579
119895
0 120579119895
1 120579
119895
119868119879
Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below
(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)
(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880
119895+1
1 119880119895+1
2 119880
119895+1
119868minus1 120585119895+1
1 120585119895+1
2 120585
119895+1
119868minus1 and
120579119895+1
1 120579119895+1
2 120579
119895+1
119868minus1 from (29)ndash(31)
(c) Calculate 119880119895+1
0 119880119895+1
119868 120585119895+1
0 120585119895+1
119868 and 120579
119895+1
0 120579119895+1
119868
from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)
(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1
1 119908119895+1
2
119908119895+1
119868minus1 from (32)
(e) Calculate 119908119895+1
0 119908119895+1
119868 from (35) and calculate w119895+1
combined with step (d)
43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box
beams and boundary conditions without shear lag effects canbe deduced as
11986312058510158401015840minus 119896sl120577 minus 119863
1
120585 = 0 (38)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (39)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868
1120579 = 0 (40)
1205851015840120575120585
10038161003816100381610038161003816
119871
0= 0 120579
1015840120575120579
10038161003816100381610038161003816
119871
0= 0 (119908
1015840minus 120579) 120575119908
10038161003816100381610038161003816
119871
0= 0 (41)
For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as
1205851015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1205851015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(42)
The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42
5 Analysis of Examples
The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864
119904= 20 times 10
5MPa 119864119888= 45 times 10
4MPa119866119904= 771 times 10
4MPa 120583119904= 028 120583
119888= 018 120588
119904= 7900 kgsdotmminus3
120588119888= 2400 kgsdotmminus3 119887
1= 1198873= 25m 119887
2= 20m 119887
4= 30m
1198875= 02m 119905
1= 1199052= 03m 119905
3= 1199055= 006m 119905
4= 009m
119871 = 30m 119897 = 03m 119899119904= 2 and 119891
119904= 300MPa
The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910
degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4
The 119896sl is calculated as [5 23 24]
119896sl =1198701
119897 (43)
6 Mathematical Problems in Engineering
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
(a) 119903 = 025
000 005 010 015 020 0250
4
8
12
16
20
24
Disp
lace
men
t at m
idsp
an (m
m)
Time (s)
(b) 119903 = 05
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(c) 119903 = 10
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(d) 119903 = 15
Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)
The spring parameter of the COMBIN14 elements can becalculated as [25]
1198701= 066119899
119904119881119906 (44)
The degree of shear connections is calculated as [5]
119903 =119899119904119881119906119871
119860119904119891119904119897 (45)
where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891
119904is the tensile strength of steel beams 119899
119904
is the number of connectors across the beam section119881119906is the
ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from
the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows
(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period
(2) The larger the degree of shear connections thesmaller the amplitude of deflection
(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams
(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects
(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear
Mathematical Problems in Engineering 7
0 1 2 3 4 5100
120
140
160
180
200
minus5 minus4 minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
Stra
in d
istrib
utio
n (10minus6)
(a) Concrete plate
0 1 2 3minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
minus220
minus230
minus240
minus250
minus260
minus270
minus280
Stra
in d
istrib
utio
n (10minus6)
(b) Steel beam bottom plate
Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)
connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection
6 Conclusions
(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams
(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams
(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect
(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)
References
[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003
[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005
[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007
[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014
[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012
[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011
[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
A998400
1A998400
2
A998400
3
d
Figure 2 Schematic of warping shape function
(2) The longitudinal displacement of any point in thetransverse section of composite box beams is assumed as thesuperposition of the longitudinal displacement based on theplain section assumption the longitudinal warping displace-ment due to the shear lag and longitudinal displacement dueto the relative interface slip It can be expressed as [5 23 24]
119906119894= 119896119888120585 minus (119911 minus 119911
119905) 120579 + 119892
119894119894 = 1 2 (7)
119906119894= 119896119904120585 minus (119911 minus 119911
119904) 120579 + 119892
119894119894 = 3 4 (8)
119896119888= minus
119860119904
1198600
119896119904=
119860119888
(1198991198600) (9)
120577 (119909 119905) = 120585 + ℎ120579 (10)
where 120579(119909 119905) is the rotation of the beam section 119860119908= 211988741199054
119860119904= 1198603+ 1198604is the cross section area of steel beams 119860
119888=
1198601+ 1198602is the cross section area of concrete slabs 120585(119909 119905) is
the longitudinal displacement difference between centroidsof the concrete slab and steel beam 120577(119909 119905) is slab-beaminterface slip 119911
119904is the 119911-coordinate of the centroid of the
steel beam 119896119904is the ratio between steel beamrsquos longitudinal
displacement due to the relative interface slip and relativeinterface slip and 119896
119888is the ratio between concrete slabrsquos
longitudinal displacement due to the relative interface slipand relative interface slip
(3) The vertical compression and transverse strain ofconcrete slabs and steel beams are ignored [5 23]
3 Vibration Differential Equation andBoundary Conditions
31 The Strain of the Cross Section The sectional strain canbe obtained from the above longitudinal displacement ofcomposite beam sections as
120576119909119894
= 119896119888
120597120585
120597119909minus (119911 minus 119911
119905)120597120579
120597119909+ 120595119894
120597119880
120597119909119894 = 1 2
120576119909119894
= 119896119904
120597120585
120597119909minus (119911 minus 119911
119904)120597120579
120597119909+ 120595119894
120597119880
120597119909119894 = 3 4
120574119909119910119894
=120597119906119894
120597119910=
120597120595119894
120597119910119880 119894 = 1 2 3
120574119909119911
=120597119908
120597119909minus 120579
(11)
where 120576119894(119894 = 1 2 3 4) are the longitudinal strain of the top
concrete slab cantilever plate bottom flange and web of steelbeams respectively 120574
119894(119894 = 1 2 3) are the shear strain of
top concrete slab cantilever plate and bottom plate of steelbeams respectively 120574
119909119911is the shear strain of the web of steel
beams 119908(119909 119905) is the vertical deflection of composite boxbeams
32 Total Potential Energy of the Composite Box Beam Thestrain energy of composite box beams is defined as [5]
119881 = 05 int
119871
0
int
1198600
(1198641199041205762+ 1198661199041205742) 119889119860119889119909
+ 05 int
119871
0
119896sl1205772119889119909 +
051198661199041198600(1199081015840minus 120579)2
120572119904
(12)
Substituting (10)-(11) into (12) gives the strain energy ofcomposite box beams as
119881 = 05 int
119871
0
[
[
11986312058510158402+ 11986511988010158402+ 2119867119880
10158401205851015840+ 11986812057910158402+ 1198691198802
minus 211987811988010158401205791015840+ 119896sl120577
2+
1198661199041198600(1199081015840minus 120579)2
120572119904
]
]
119889119909
(13)
where119863 = 1198641198881198962
119888119860119888+1198641199041198962
119904119860119904 119865 = 119864
119888119861119888119891+119864119904119861119904119891119867 = 119864
119888119861119888ℎ+
119864119904119861119904ℎ 119868 = 119864
119888119861119888119894+119864119904119861119904119894 119869 = 119866
119888119861119888119895+119866119904119861119904119895 119878 = 119864
119888119861119888119904+119864119904119861119904119904
119861119888119891
= int119860119888
1205952119889119860 119861
119904119891= int119860119904
1205952119889119860 119861
119888ℎ= int119860119888
119896119888120595119889119860 119861
119904ℎ=
int119860119904
119896119904120595119889119860 119861
119888119894= int119860119888
(119911 minus 119911119905)2119889119860 119861
119904119894= int119860119904
(119911 minus 119911119904)2119889119860 119861
119888119895=
int119860119888
(120597120595120597119910)2119889119860 119861
119904119895= int1198603
(120597120595120597119910)2119889119860 119861
119888119904= int119860119888
(119911minus119911119905)120595 119889119860
119861119904119904
= int119860119904
(119911 minus 119911119904)120595 119889119860 and 120572
119904is the correction coefficient of
shear deformation Considering that the webs bear most ofthe vertical shear force in section here the value of 120572
119904is taken
as 1198600(211988741199054) and 2119887
41199054is the section area of webs 119896sl is the
slip stiffness between the concrete slab and the steel beam119871 isthe span of the composite box beam 119866
119904is the shear modulus
of the steel beamThe kinetic energy of the composite box beam is [5]
119879 =1
2int
119871
0
int
119860
(1205882+ 119898
2) 119889119860119889119909 (14)
Substituting of (7) and (8) yields
119879
=1
2int
119871
(1198982+ 1198631
1205852+ 11986512+ 21198671
120585 + 1198681
1205792minus 21198781 120579) 119889119909
(15)
where1198631= 1205881198881198962
119888119860119888+1205881199041198962
1199041198601199041198651= 120588119888119861119888119891+1205881199041198611199041198911198671= 120588119888119861119888ℎ+
120588119904119861119904ℎ 1198681= 120588119888119861119888119894+ 120588119904119861119904119894 1198781= 120588119888119861119888119904+ 120588119904119861119904119904119898 = 120588
119888119860119888+ 120588119904119860119904
119860 = 119860119904+ 119860119888 and 120588
119888and 120588119904are the density of concrete slabs
and steel beams respectively
4 Mathematical Problems in Engineering
The work done by the external loads can be expressed as
119882 = int
119871
119902 (119909 119905) 119908 119889119909 (16)
where 119902(119909 119905) is the distribution function of arbitrary load
33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as
11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579
10158401015840minus 1198651 minus 119867
1
120585 + 1198781
120579 = 0 (17)
11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867
1 minus 119863
1
120585 = 0 (18)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (19)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878
1 minus 1198681
120579 = 0 (20)
(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880
10038161003816100381610038161003816
119871
0= 0 (21)
(1198631205851015840+ 119867119880
1015840) 120575120585
10038161003816100381610038161003816
119871
0= 0 (22)
1198661199041198600(1199081015840minus 120579)
120572119904
120575119908
10038161003816100381610038161003816100381610038161003816100381610038161003816
119871
0
= 0 (23)
(1198681205791015840minus 1198781198801015840) 120575120579
10038161003816100381610038161003816
119871
0= 0 (24)
Taking a simply supported beam as example (21)ndash(24)give boundary conditions as
1198801015840(119871 119905) = 120585
1015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1198801015840(0 119905) = 120585
1015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(25)
given the initial conditions as
119880 (119909 0) = 1198800(119909) (119909 0) = 119880
0(119909)
120585 (119909 0) = 1205850(119909)
120585 (119909 0) = 120585
0(119909)
119908 (119909 0) = 1199080(119909) (119909 0) = 119908
0(119909)
120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579
0(119909)
(26)
4 Finite Difference Method ofVibration Differential Equation
41 Difference Scheme Let solution domain be 120590 = (119909 119905) |
0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that
119909119894= 119894120592 (119894 = 0 1 2 119868)
119905119895= 119895120591 (119895 = 0 1 2 119869)
(27)
where 119868 = 119871120592 119869 = 119879120591
Let
119880119895
119894= 119880 (119909
119894 119905119895) 120585
119895
119894= 120585 (119909
119894 119905119895)
119908119895
119894= 119908 (119909
119894 119905119895) 120579
119895
119894= 120579 (119909
119894 119905119895)
(28)
The central difference calculation of governing differen-tial equation yields
119865
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119867
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119878
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 1198651
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 119869119880119895
119894minus 1198671
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
+ 1198781
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(29)
119867
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119863
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119896sl (120585119895
119894+ ℎ120579119895
119894) minus 119867
1
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198631
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(30)
1198661199041198600
120572119904
(
119908119895
119894+1minus 119908119895
119894minus1
2120592minus 120579119895
119894) + 119868
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 119878
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 1198781
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198681
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
minus 119896slℎ (120585119895
119894+ ℎ120579119895
119894) = 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(31)
1198661199041198600
120572119904
(
119908119895
119894+1minus 2119908119895
119894+ 119908119895
119894minus1
1205922
minus
120579119895
119894+1minus 120579119895
119894minus1
2120592)
+ 119902119895
119894minus 119898
119908119895+1
119894minus 2119908119895
119894+ 119908119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(32)
Mathematical Problems in Engineering 5
The difference calculation of boundary conditions yields
119880119895
119868minus 119880119895
119868minus1
120592=
119880119895
1minus 119880119895
0
120592= 0 119895 = 0 1 119869
(33)
120585119895
119868minus 120585119895
119868minus1
120592=
120585119895
1minus 120585119895
0
120592= 0 119895 = 0 1 119869
(34)
119908119895
119868= 119908119895
0= 0 119895 = 0 1 119869 (35)
120579119895
119868minus 120579119895
119868minus1
120592=
120579119895
1minus 120579119895
0
120592= 0 119895 = 0 1 119869
(36)
The difference calculation of initial condition yields
1198800
119894= 1198800119894
1198801
119894minus 1198800
119894
120591= 1198800119894
119894 = 0 1 2 119868
1205850
119894= 1205850119894
1205851
119894minus 1205850
119894
120591= 1205850119894
119894 = 0 1 2 119868
1199080
119894= 1199080119894
1199081
119894minus 1199080
119894
120591= 1199080119894
119894 = 0 1 2 119868
1205790
119894= 1205790119894
1205791
119894minus 1205790
119894
120591= 1205790119894
119894 = 0 1 2 119868
(37)
42 Solution Step Let U119895 = 119880119895
0 119880119895
1 119880
119895
119868119879 120585119895 = 120585
119895
0 120585119895
1
120585119895
119868119879 w119895 = 119908
119895
0 119908119895
1 119908
119895
119868119879 and 120579119895 = 120579
119895
0 120579119895
1 120579
119895
119868119879
Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below
(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)
(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880
119895+1
1 119880119895+1
2 119880
119895+1
119868minus1 120585119895+1
1 120585119895+1
2 120585
119895+1
119868minus1 and
120579119895+1
1 120579119895+1
2 120579
119895+1
119868minus1 from (29)ndash(31)
(c) Calculate 119880119895+1
0 119880119895+1
119868 120585119895+1
0 120585119895+1
119868 and 120579
119895+1
0 120579119895+1
119868
from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)
(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1
1 119908119895+1
2
119908119895+1
119868minus1 from (32)
(e) Calculate 119908119895+1
0 119908119895+1
119868 from (35) and calculate w119895+1
combined with step (d)
43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box
beams and boundary conditions without shear lag effects canbe deduced as
11986312058510158401015840minus 119896sl120577 minus 119863
1
120585 = 0 (38)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (39)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868
1120579 = 0 (40)
1205851015840120575120585
10038161003816100381610038161003816
119871
0= 0 120579
1015840120575120579
10038161003816100381610038161003816
119871
0= 0 (119908
1015840minus 120579) 120575119908
10038161003816100381610038161003816
119871
0= 0 (41)
For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as
1205851015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1205851015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(42)
The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42
5 Analysis of Examples
The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864
119904= 20 times 10
5MPa 119864119888= 45 times 10
4MPa119866119904= 771 times 10
4MPa 120583119904= 028 120583
119888= 018 120588
119904= 7900 kgsdotmminus3
120588119888= 2400 kgsdotmminus3 119887
1= 1198873= 25m 119887
2= 20m 119887
4= 30m
1198875= 02m 119905
1= 1199052= 03m 119905
3= 1199055= 006m 119905
4= 009m
119871 = 30m 119897 = 03m 119899119904= 2 and 119891
119904= 300MPa
The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910
degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4
The 119896sl is calculated as [5 23 24]
119896sl =1198701
119897 (43)
6 Mathematical Problems in Engineering
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
(a) 119903 = 025
000 005 010 015 020 0250
4
8
12
16
20
24
Disp
lace
men
t at m
idsp
an (m
m)
Time (s)
(b) 119903 = 05
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(c) 119903 = 10
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(d) 119903 = 15
Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)
The spring parameter of the COMBIN14 elements can becalculated as [25]
1198701= 066119899
119904119881119906 (44)
The degree of shear connections is calculated as [5]
119903 =119899119904119881119906119871
119860119904119891119904119897 (45)
where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891
119904is the tensile strength of steel beams 119899
119904
is the number of connectors across the beam section119881119906is the
ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from
the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows
(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period
(2) The larger the degree of shear connections thesmaller the amplitude of deflection
(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams
(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects
(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear
Mathematical Problems in Engineering 7
0 1 2 3 4 5100
120
140
160
180
200
minus5 minus4 minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
Stra
in d
istrib
utio
n (10minus6)
(a) Concrete plate
0 1 2 3minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
minus220
minus230
minus240
minus250
minus260
minus270
minus280
Stra
in d
istrib
utio
n (10minus6)
(b) Steel beam bottom plate
Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)
connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection
6 Conclusions
(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams
(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams
(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect
(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)
References
[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003
[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005
[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007
[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014
[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012
[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011
[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
The work done by the external loads can be expressed as
119882 = int
119871
119902 (119909 119905) 119908 119889119909 (16)
where 119902(119909 119905) is the distribution function of arbitrary load
33 VibrationDifferential Equation and Boundary ConditionsThegoverning equations of vibration of composite box beamsand corresponding boundary conditions can be deducedbased on Hamilton principle as
11986511988010158401015840+ 11986712058510158401015840minus 119869119880 minus 119878120579
10158401015840minus 1198651 minus 119867
1
120585 + 1198781
120579 = 0 (17)
11986711988010158401015840+ 11986312058510158401015840minus 119896sl120577 minus 119867
1 minus 119863
1
120585 = 0 (18)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (19)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 11987811988010158401015840minus 119896sl120577ℎ + 119878
1 minus 1198681
120579 = 0 (20)
(1198651198801015840+ 1198671205851015840minus 1198781205791015840) 120575119880
10038161003816100381610038161003816
119871
0= 0 (21)
(1198631205851015840+ 119867119880
1015840) 120575120585
10038161003816100381610038161003816
119871
0= 0 (22)
1198661199041198600(1199081015840minus 120579)
120572119904
120575119908
10038161003816100381610038161003816100381610038161003816100381610038161003816
119871
0
= 0 (23)
(1198681205791015840minus 1198781198801015840) 120575120579
10038161003816100381610038161003816
119871
0= 0 (24)
Taking a simply supported beam as example (21)ndash(24)give boundary conditions as
1198801015840(119871 119905) = 120585
1015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1198801015840(0 119905) = 120585
1015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(25)
given the initial conditions as
119880 (119909 0) = 1198800(119909) (119909 0) = 119880
0(119909)
120585 (119909 0) = 1205850(119909)
120585 (119909 0) = 120585
0(119909)
119908 (119909 0) = 1199080(119909) (119909 0) = 119908
0(119909)
120579 (119909 0) = 1205790(119909) 120579 (119909 0) = 120579
0(119909)
(26)
4 Finite Difference Method ofVibration Differential Equation
41 Difference Scheme Let solution domain be 120590 = (119909 119905) |
0 le 119909 le 119871 0 le 119905 le 119879 119879 is the end time of solution arectangularmesh ismade in the solution area with a time stepof 120591 and space step of 120592 so that
119909119894= 119894120592 (119894 = 0 1 2 119868)
119905119895= 119895120591 (119895 = 0 1 2 119869)
(27)
where 119868 = 119871120592 119869 = 119879120591
Let
119880119895
119894= 119880 (119909
119894 119905119895) 120585
119895
119894= 120585 (119909
119894 119905119895)
119908119895
119894= 119908 (119909
119894 119905119895) 120579
119895
119894= 120579 (119909
119894 119905119895)
(28)
The central difference calculation of governing differen-tial equation yields
119865
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119867
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119878
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 1198651
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 119869119880119895
119894minus 1198671
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
+ 1198781
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(29)
119867
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 119863
120585119895
119894+1minus 2120585119895
119894+ 120585119895
119894minus1
1205922
minus 119896sl (120585119895
119894+ ℎ120579119895
119894) minus 119867
1
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198631
120585119895+1
119894minus 2120585119895
119894+ 120585119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(30)
1198661199041198600
120572119904
(
119908119895
119894+1minus 119908119895
119894minus1
2120592minus 120579119895
119894) + 119868
120579119895
119894+1minus 2120579119895
119894+ 120579119895
119894minus1
1205922
minus 119878
119880119895
119894+1minus 2119880119895
119894+ 119880119895
119894minus1
1205922
+ 1198781
119880119895+1
119894minus 2119880119895
119894+ 119880119895minus1
119894
1205912
minus 1198681
120579119895+1
119894minus 2120579119895
119894+ 120579119895minus1
119894
1205912
minus 119896slℎ (120585119895
119894+ ℎ120579119895
119894) = 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(31)
1198661199041198600
120572119904
(
119908119895
119894+1minus 2119908119895
119894+ 119908119895
119894minus1
1205922
minus
120579119895
119894+1minus 120579119895
119894minus1
2120592)
+ 119902119895
119894minus 119898
119908119895+1
119894minus 2119908119895
119894+ 119908119895minus1
119894
1205912
= 0
(119894 = 1 119868 minus 1 119895 = 1 2 119869)
(32)
Mathematical Problems in Engineering 5
The difference calculation of boundary conditions yields
119880119895
119868minus 119880119895
119868minus1
120592=
119880119895
1minus 119880119895
0
120592= 0 119895 = 0 1 119869
(33)
120585119895
119868minus 120585119895
119868minus1
120592=
120585119895
1minus 120585119895
0
120592= 0 119895 = 0 1 119869
(34)
119908119895
119868= 119908119895
0= 0 119895 = 0 1 119869 (35)
120579119895
119868minus 120579119895
119868minus1
120592=
120579119895
1minus 120579119895
0
120592= 0 119895 = 0 1 119869
(36)
The difference calculation of initial condition yields
1198800
119894= 1198800119894
1198801
119894minus 1198800
119894
120591= 1198800119894
119894 = 0 1 2 119868
1205850
119894= 1205850119894
1205851
119894minus 1205850
119894
120591= 1205850119894
119894 = 0 1 2 119868
1199080
119894= 1199080119894
1199081
119894minus 1199080
119894
120591= 1199080119894
119894 = 0 1 2 119868
1205790
119894= 1205790119894
1205791
119894minus 1205790
119894
120591= 1205790119894
119894 = 0 1 2 119868
(37)
42 Solution Step Let U119895 = 119880119895
0 119880119895
1 119880
119895
119868119879 120585119895 = 120585
119895
0 120585119895
1
120585119895
119868119879 w119895 = 119908
119895
0 119908119895
1 119908
119895
119868119879 and 120579119895 = 120579
119895
0 120579119895
1 120579
119895
119868119879
Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below
(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)
(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880
119895+1
1 119880119895+1
2 119880
119895+1
119868minus1 120585119895+1
1 120585119895+1
2 120585
119895+1
119868minus1 and
120579119895+1
1 120579119895+1
2 120579
119895+1
119868minus1 from (29)ndash(31)
(c) Calculate 119880119895+1
0 119880119895+1
119868 120585119895+1
0 120585119895+1
119868 and 120579
119895+1
0 120579119895+1
119868
from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)
(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1
1 119908119895+1
2
119908119895+1
119868minus1 from (32)
(e) Calculate 119908119895+1
0 119908119895+1
119868 from (35) and calculate w119895+1
combined with step (d)
43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box
beams and boundary conditions without shear lag effects canbe deduced as
11986312058510158401015840minus 119896sl120577 minus 119863
1
120585 = 0 (38)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (39)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868
1120579 = 0 (40)
1205851015840120575120585
10038161003816100381610038161003816
119871
0= 0 120579
1015840120575120579
10038161003816100381610038161003816
119871
0= 0 (119908
1015840minus 120579) 120575119908
10038161003816100381610038161003816
119871
0= 0 (41)
For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as
1205851015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1205851015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(42)
The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42
5 Analysis of Examples
The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864
119904= 20 times 10
5MPa 119864119888= 45 times 10
4MPa119866119904= 771 times 10
4MPa 120583119904= 028 120583
119888= 018 120588
119904= 7900 kgsdotmminus3
120588119888= 2400 kgsdotmminus3 119887
1= 1198873= 25m 119887
2= 20m 119887
4= 30m
1198875= 02m 119905
1= 1199052= 03m 119905
3= 1199055= 006m 119905
4= 009m
119871 = 30m 119897 = 03m 119899119904= 2 and 119891
119904= 300MPa
The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910
degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4
The 119896sl is calculated as [5 23 24]
119896sl =1198701
119897 (43)
6 Mathematical Problems in Engineering
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
(a) 119903 = 025
000 005 010 015 020 0250
4
8
12
16
20
24
Disp
lace
men
t at m
idsp
an (m
m)
Time (s)
(b) 119903 = 05
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(c) 119903 = 10
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(d) 119903 = 15
Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)
The spring parameter of the COMBIN14 elements can becalculated as [25]
1198701= 066119899
119904119881119906 (44)
The degree of shear connections is calculated as [5]
119903 =119899119904119881119906119871
119860119904119891119904119897 (45)
where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891
119904is the tensile strength of steel beams 119899
119904
is the number of connectors across the beam section119881119906is the
ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from
the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows
(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period
(2) The larger the degree of shear connections thesmaller the amplitude of deflection
(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams
(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects
(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear
Mathematical Problems in Engineering 7
0 1 2 3 4 5100
120
140
160
180
200
minus5 minus4 minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
Stra
in d
istrib
utio
n (10minus6)
(a) Concrete plate
0 1 2 3minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
minus220
minus230
minus240
minus250
minus260
minus270
minus280
Stra
in d
istrib
utio
n (10minus6)
(b) Steel beam bottom plate
Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)
connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection
6 Conclusions
(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams
(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams
(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect
(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)
References
[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003
[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005
[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007
[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014
[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012
[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011
[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
The difference calculation of boundary conditions yields
119880119895
119868minus 119880119895
119868minus1
120592=
119880119895
1minus 119880119895
0
120592= 0 119895 = 0 1 119869
(33)
120585119895
119868minus 120585119895
119868minus1
120592=
120585119895
1minus 120585119895
0
120592= 0 119895 = 0 1 119869
(34)
119908119895
119868= 119908119895
0= 0 119895 = 0 1 119869 (35)
120579119895
119868minus 120579119895
119868minus1
120592=
120579119895
1minus 120579119895
0
120592= 0 119895 = 0 1 119869
(36)
The difference calculation of initial condition yields
1198800
119894= 1198800119894
1198801
119894minus 1198800
119894
120591= 1198800119894
119894 = 0 1 2 119868
1205850
119894= 1205850119894
1205851
119894minus 1205850
119894
120591= 1205850119894
119894 = 0 1 2 119868
1199080
119894= 1199080119894
1199081
119894minus 1199080
119894
120591= 1199080119894
119894 = 0 1 2 119868
1205790
119894= 1205790119894
1205791
119894minus 1205790
119894
120591= 1205790119894
119894 = 0 1 2 119868
(37)
42 Solution Step Let U119895 = 119880119895
0 119880119895
1 119880
119895
119868119879 120585119895 = 120585
119895
0 120585119895
1
120585119895
119868119879 w119895 = 119908
119895
0 119908119895
1 119908
119895
119868119879 and 120579119895 = 120579
119895
0 120579119895
1 120579
119895
119868119879
Given U119895minus1 120585119895minus1 w119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 the solving ofU119895+1 120585119895+1 w119895+1 and 120579119895+1 follows the steps below
(a) Calculate U0 1205850 w0 1205790 U1 1205851 w1 and 1205791 from theinitial conditions given in (37)
(b) Given U119895minus1 120585119895minus1 120579119895minus1 U119895 120585119895 w119895 and 120579119895 calcu-late 119880
119895+1
1 119880119895+1
2 119880
119895+1
119868minus1 120585119895+1
1 120585119895+1
2 120585
119895+1
119868minus1 and
120579119895+1
1 120579119895+1
2 120579
119895+1
119868minus1 from (29)ndash(31)
(c) Calculate 119880119895+1
0 119880119895+1
119868 120585119895+1
0 120585119895+1
119868 and 120579
119895+1
0 120579119895+1
119868
from (33) (34) and (36) and the solution of U119895+1120585119895+1 and 120579119895+1 is obtained in combination withstep (b)
(d) Given w119895minus1 w119895 and 120579119895 determine 119908119895+1
1 119908119895+1
2
119908119895+1
119868minus1 from (32)
(e) Calculate 119908119895+1
0 119908119895+1
119868 from (35) and calculate w119895+1
combined with step (d)
43 Degeneration of theVibrationDifferential Equation Like-wise the governing equations of vibration of composite box
beams and boundary conditions without shear lag effects canbe deduced as
11986312058510158401015840minus 119896sl120577 minus 119863
1
120585 = 0 (38)
1198661199041198600(11990810158401015840minus 1205791015840)
120572119904
minus 119898 + 119902 (119909 119905) = 0 (39)
1198661199041198600(1199081015840minus 120579)
120572119904
+ 11986812057910158401015840minus 119896sl120577ℎ minus 119868
1120579 = 0 (40)
1205851015840120575120585
10038161003816100381610038161003816
119871
0= 0 120579
1015840120575120579
10038161003816100381610038161003816
119871
0= 0 (119908
1015840minus 120579) 120575119908
10038161003816100381610038161003816
119871
0= 0 (41)
For beamswith two ends simply supported the boundaryconditions can be expressed from (41) as
1205851015840(119871 119905) = 120579
1015840(119871 119905) = 119908 (119871 119905) = 0
1205851015840(0 119905) = 120579
1015840(0 119905) = 119908 (0 119905) = 0
(42)
The vibration differential equation of composite boxbeams without considering shear lag effects is a degenera-tion of the one considering the shear lag effects the solutionmethod of which can be referred to in Sections 41 and 42
5 Analysis of Examples
The validity of the proposed method is verified by compar-ing to numerical results from finite element method Thecomparisons are made on four simply supported compositebox beams with different degree of shear connections undersuddenly imposed distributed loads The dynamic responsesof beams with and without shear lag effects are analyzedThe distributed load is taken as 119902 = 300 kNm with a timestep of 120591 = 000002 s and a space step of 120592 = 03m Themechanical and geometrical parameters of composite boxbeams are taken as 119864
119904= 20 times 10
5MPa 119864119888= 45 times 10
4MPa119866119904= 771 times 10
4MPa 120583119904= 028 120583
119888= 018 120588
119904= 7900 kgsdotmminus3
120588119888= 2400 kgsdotmminus3 119887
1= 1198873= 25m 119887
2= 20m 119887
4= 30m
1198875= 02m 119905
1= 1199052= 03m 119905
3= 1199055= 006m 119905
4= 009m
119871 = 30m 119897 = 03m 119899119904= 2 and 119891
119904= 300MPa
The commercial finite element software ANSYS is used inthis study In the finite element model the concrete slab andsteel beam are modeled by SOLID65 and SHELL43 elementsrespectively Shear connector is modeled by COMBIN14elements being spring elements [25] The aspect ratio of themesh is kept close to one throughout the mean mesh sizevaries from 02m to 03m and about 9200 finite elementsin total are employed The transverse distributed loads areimposed on the top flanges The translational 119909 and 119910
degrees of freedom of all nodes at two ends of beams arerestrained The torsion at the beam ends is thus restrainedThe translational 119911 DOF of the left end is restrained to meetthe static determined requirement and allow rotation alongthe moment The results are shown in Figures 3 and 4
The 119896sl is calculated as [5 23 24]
119896sl =1198701
119897 (43)
6 Mathematical Problems in Engineering
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
(a) 119903 = 025
000 005 010 015 020 0250
4
8
12
16
20
24
Disp
lace
men
t at m
idsp
an (m
m)
Time (s)
(b) 119903 = 05
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(c) 119903 = 10
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(d) 119903 = 15
Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)
The spring parameter of the COMBIN14 elements can becalculated as [25]
1198701= 066119899
119904119881119906 (44)
The degree of shear connections is calculated as [5]
119903 =119899119904119881119906119871
119860119904119891119904119897 (45)
where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891
119904is the tensile strength of steel beams 119899
119904
is the number of connectors across the beam section119881119906is the
ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from
the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows
(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period
(2) The larger the degree of shear connections thesmaller the amplitude of deflection
(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams
(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects
(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear
Mathematical Problems in Engineering 7
0 1 2 3 4 5100
120
140
160
180
200
minus5 minus4 minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
Stra
in d
istrib
utio
n (10minus6)
(a) Concrete plate
0 1 2 3minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
minus220
minus230
minus240
minus250
minus260
minus270
minus280
Stra
in d
istrib
utio
n (10minus6)
(b) Steel beam bottom plate
Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)
connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection
6 Conclusions
(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams
(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams
(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect
(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)
References
[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003
[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005
[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007
[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014
[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012
[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011
[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
(a) 119903 = 025
000 005 010 015 020 0250
4
8
12
16
20
24
Disp
lace
men
t at m
idsp
an (m
m)
Time (s)
(b) 119903 = 05
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(c) 119903 = 10
000 005 010 015 020 0250
4
8
12
16
20
24
Time (s)
Disp
lace
men
t at m
idsp
an (m
m)
ANSYSThis paper 120582 = 1
This paper 120582 = 0
(d) 119903 = 15
Figure 3 Displacement time history at midspan of composite beams (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion ofshear lag)
The spring parameter of the COMBIN14 elements can becalculated as [25]
1198701= 066119899
119904119881119906 (44)
The degree of shear connections is calculated as [5]
119903 =119899119904119881119906119871
119860119904119891119904119897 (45)
where 119897 is the spacing of shear connectors 119903 is the degree ofshear connections 119891
119904is the tensile strength of steel beams 119899
119904
is the number of connectors across the beam section119881119906is the
ultimate shear strength of a single shear connectorFrom Figures 3 and 4 it can be seen that the results from
the proposed method agree well with those from ANSYSfor different degrees of shear connections This verifies theaccuracy of the proposed method and some meaningfulconclusions for engineering design can be drawn as follows
(1) For various degrees of shear connections the vibra-tion period of composite beams with and withoutshear lag effects is almost the same which indicatesthat the shear lag and shear connection degree havelittle impact on the vibration period
(2) The larger the degree of shear connections thesmaller the amplitude of deflection
(3) The influence of the shear lag on the amplitude ofdeflection is not significant indicating that it has littleeffect on the deflection of beams
(4) The longitudinal strain in the concrete slab and steelbeam bottom plate shows obvious shear lag effects
(5) The longitudinal strain in the concrete slab reducesobviously with an increment in the degree of shear
Mathematical Problems in Engineering 7
0 1 2 3 4 5100
120
140
160
180
200
minus5 minus4 minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
Stra
in d
istrib
utio
n (10minus6)
(a) Concrete plate
0 1 2 3minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
minus220
minus230
minus240
minus250
minus260
minus270
minus280
Stra
in d
istrib
utio
n (10minus6)
(b) Steel beam bottom plate
Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)
connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection
6 Conclusions
(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams
(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams
(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect
(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)
References
[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003
[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005
[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007
[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014
[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012
[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011
[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
0 1 2 3 4 5100
120
140
160
180
200
minus5 minus4 minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
Stra
in d
istrib
utio
n (10minus6)
(a) Concrete plate
0 1 2 3minus3 minus2 minus1
ANSYS r = 025
This paper 120582 = 1 r = 025
This paper 120582 = 0 r = 025ANSYS r = 05
This paper 120582 = 1 r = 05
This paper 120582 = 0 r = 05ANSYS r = 1
This paper 120582 = 1 r = 1
This paper 120582 = 0 r = 1
Y-coordinate (m)
minus220
minus230
minus240
minus250
minus260
minus270
minus280
Stra
in d
istrib
utio
n (10minus6)
(b) Steel beam bottom plate
Figure 4 Strain distribution in the flanges of midspan section (120582 = 1 represents inclusion of shear lag and 120582 = 0 is for exclusion of shear lag)
connections which indicates that the shear connec-tion degree has a great effect on the longitudinal strainof concrete slab Nevertheless the longitudinal strainin the steel beam bottom plate changes little with anincrement in the degree of shear connection
6 Conclusions
(1) Numerical analyses are carried out to verify theaccuracy of the proposed theory The comprehensiveconsiderations of shear lag shear deformation andinterface slip yield an accurate prediction of dynamicresponses of composite box beams
(2) The degree of shear connections has little impacton the vibration period of composite beams but hassignificant impact on the amplitude of deflection ofcomposite beams Nevertheless the shear lag effecthas limited contribution to the deflection or thevibration period of composite beams
(3) The longitudinal strain in the concrete slabs andbottom plate of steel beams shows an obvious shearlag effect
(4) The longitudinal strain in the concrete slabs reducesgreatly with increasing shear connection degree Nev-ertheless the longitudinal strain in the steel beambottom plate changes little with an increment in thedegree of shear connection
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Function of China(51408449 51378502) and the Fundamental Research Fundsfor the Central Universities of China (2014-IV-049)
References
[1] J Nie and C S Cai ldquoSteel-concrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003
[2] F-F Sun and O S Bursi ldquoDisplacement-based and two-fieldmixed variational formulations for composite beams with shearlagrdquo Journal of Engineering Mechanics vol 131 no 2 pp 199ndash210 2005
[3] G Ranzi and A Zona ldquoA steel-concrete composite beammodelwith partial interaction including the shear deformability ofthe steel componentrdquo Engineering Structures vol 29 no 11 pp3026ndash3041 2007
[4] Z Wangbao J Lizhong K Juntao and B Minxi ldquoDistortionalbuckling analysis of steel-concrete composite girders in negativemoment areardquoMathematical Problems in Engineering vol 2014Article ID 635617 10 pages 2014
[5] W-B Zhou L-Z Jiang Z-J Liu and X-J Liu ldquoClosed-form solution for shear lag effects of steel-concrete compositebox beams considering shear deformation and sliprdquo Journal ofCentral South University vol 19 no 10 pp 2976ndash2982 2012
[6] F Gara G Ranzi and G Leoni ldquoSimplified method of analysisaccounting for shear-lag effects in composite bridge decksrdquoJournal of Constructional Steel Research vol 67 no 10 pp 1684ndash1697 2011
[7] F Gara G Leoni and L Dezi ldquoA beam finite element includingshear lag effect for the time-dependent analysis of steel-concrete
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
composite decksrdquoEngineering Structures vol 31 no 8 pp 1888ndash1902 2009
[8] A Morassi and L Rocchetto ldquoA damage analysis of steel-concrete composite beams via dynamic methods part I Exper-imental resultsrdquo Journal of Vibration and Control vol 9 no 5pp 507ndash527 2003
[9] M Dilena and A Morassi ldquoA damage analysis of steel-concretecomposite beams via dynamic methods part II Analyticalmodels anddamage detectionrdquo Journal of Vibration andControlvol 9 no 5 pp 529ndash565 2003
[10] M Dilena and A Morassi ldquoVibrations of steelmdashconcretecomposite beamswith partially degraded connection and appli-cations to damage detectionrdquo Journal of Sound and Vibrationvol 320 no 1-2 pp 101ndash124 2009
[11] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997
[12] G Biscontin A Morassi and P Wendel ldquoVibrations of steel-concrete composite beamsrdquo Journal of Vibration and Controlvol 6 no 5 pp 691ndash714 2000
[13] S Berczynski and T Wroblewski ldquoVibration of steel-concretecomposite beams using the Timoshenko beam modelrdquo Journalof Vibration and Control vol 11 no 6 pp 829ndash848 2005
[14] S Berczynski and T Wroblewski ldquoExperimental verification ofnatural vibration models of steel-concrete composite beamsrdquoJournal of Vibration and Control vol 16 no 14 pp 2057ndash20812010
[15] R Xu and Y Wu ldquoStatic dynamic and buckling analysisof partial interaction composite members using Timoshenkorsquosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007
[16] X Shen W Chen Y Wu and R Xu ldquoDynamic analysis ofpartial-interaction composite beamsrdquo Composites Science andTechnology vol 71 no 10 pp 1286ndash1294 2011
[17] Z Shen and H Zhong ldquoStatic and vibrational analysis ofpartially composite beams using the weak-form quadratureelement methodrdquo Mathematical Problems in Engineering vol2012 Article ID 974023 23 pages 2012
[18] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoDynamic response of composite beams with partial shearinteraction using a higher-order beam theoryrdquo Journal ofStructural Engineering vol 139 no 1 pp 47ndash56 2013
[19] J G S da Silva S A L de Andrade and E D C LopesldquoParametric modelling of the dynamic behaviour of a steel-concrete composite floorrdquo Engineering Structures vol 75 pp327ndash339 2014
[20] S Lenci and J Warminski ldquoFree and forced nonlinear oscil-lations of a two-layer composite beam with interface sliprdquoNonlinear Dynamics vol 70 no 3 pp 2071ndash2087 2012
[21] Q-H Nguyen M Hjiaj and P Le Grognec ldquoAnalyticalapproach for free vibration analysis of two-layer Timoshenkobeams with interlayer sliprdquo Journal of Sound and Vibration vol331 no 12 pp 2949ndash2961 2012
[22] W-A Wang Q Li C-H Zhao and W-L Zhuang ldquoDynamicproperties of long-span steel-concrete composite bridges withexternal tendonsrdquo Journal of Highway and TransportationResearch andDevelopment (English Edition) vol 7 no 4 pp 30ndash38 2013
[23] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013
[24] W-B Zhou L-Z Jiang Z-W Yu and Z Huang ldquoFree vibra-tion characteristics of steel-concrete composite continuous boxgirder considering shear lag and sliprdquo China Journal of Highwayand Transport vol 26 no 5 pp 88ndash94 2013
[25] J Nie J Fan and C S Cai ldquoStiffness and deflection of steel-concrete composite beams under negative bendingrdquo Journal ofStructural Engineering vol 130 no 11 pp 1842ndash1851 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of